nLab Mather-Yau theorem

Statement

The isomorphism type of the local ring of an isolated complex hypersurface singularity? is determined by its Tjurina algebra. In positive characteristic there is an analogue where the local ring is determined by a higher Tjurina algebra instead.

Literature

The result is proven in

• John N. Mather, Stephen S.-T. Yau, Criterion for biholomorphic equivalence of isolated hypersurface singularities, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 10, Phys. Sci., 5946–5947 https://mathscinet.ams.org/mathscinet-getitem?mr=773820 doi

Some extensions and analogues include

• Gert-Martin Greuel, Thuy Huong Pham, Mather-Yau theorem in positive characteristic, J. Algebraic Geom. 26 (2017), 347-355 doi
• Z. Hua, G. Zhou, Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras, Math. Ann. (2022) doi arXiv:1803.06128

category: algebraic geometry